Basic statistics for Green Belts
Statistics at the Green Belt level focuses on describing processes and detecting when they change. You need to understand measures of central tendency, variation, and the normal distribution well enough to calculate, interpret, and connect them to process capability and control charts.
Measures of central tendency
| Measure | Definition | Use when |
|---|---|---|
| Mean (x̅) | Sum of all values ÷ n. Sensitive to outliers. | Data is symmetric, no extreme outliers |
| Median | Middle value when sorted. Resistant to outliers. | Skewed data, ordinal data, outliers present |
| Mode | Most frequent value | Categorical data; bimodal distributions |
Measures of variation
| Measure | Formula (sample) | Practical meaning |
|---|---|---|
| Range (R) | Max − Min | Simple; used in X̅-R charts. Sensitive to outliers. |
| Variance (s²) | Σ(x−x̅)² ÷ (n−1) | Average squared deviation; not in original units |
| Standard deviation (s) | √variance | Typical distance of values from the mean; same units as data |
Five fill weights (mg): 248, 250, 251, 249, 252
Normal distribution and the empirical rule
Many natural processes follow a normal (bell-shaped) distribution. If a process is normal and in control:
| Interval | % of data within | Practical use |
|---|---|---|
| Mean ± 1σ | ~68.3% | Most of the common variation |
| Mean ± 2σ | ~95.4% | Sometimes used as “warning” limits (not the standard 3σ control limit) |
| Mean ± 3σ | ~99.73% | Standard control chart limits; 0.27% false alarm rate |
Note: control chart limits at ±3σ are chosen to minimize false signals, not to cover 99.73% for its own sake.
Population vs. sample statistics
| Population | Sample | |
|---|---|---|
| Mean | μ (mu) | x̅ (x-bar) |
| Std dev | σ (sigma) | s |
| Denominator for variance | N | n−1 (Bessel’s correction) |
In Six Sigma, processes are sampled — use s and x̅ for data you have collected. Use μ and σ when describing the theoretical process distribution or when process parameters are known.
Skewness and what it tells you
- Right-skewed (positive skew): Tail extends to the right; mean > median. Common in cycle times, costs — a few very long events pull the mean up.
- Left-skewed (negative skew): Tail extends to the left; mean < median. Common when there is a natural upper limit (e.g., purity cannot exceed 100%).
- Symmetric: Mean ≈ median ≈ mode; no transformation needed for most analyses.
Common exam traps
- Use n−1 for sample standard deviation. Dividing by n underestimates variance in a sample.
- Standard deviation is in the same units as the data. Variance is in squared units — not directly interpretable in the original scale.
- The empirical rule applies only to approximately normal data. Skewed or bimodal data does not follow the 68–95–99.7 rule.
- Range is fast but weak. Range uses only two data points; standard deviation uses all data. Range charts are used in control charts for practical reasons, not because they are more precise.
The free 20-question CSSGB diagnostic covers all six Body of Knowledge sections, with an explanation for every answer.
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